A little tip to better your backstroke, is to think of using the shoulder and not to think about the arm or hand for arm recovery . Think of shrugging or popping the shoulder up as you begin recovery then and let the shoulder brush lightly against your face as it passes by . It will keep you more in rhythm. The shoulder is an extension of motion that is set up by the hips and core .This tip definitely help my backstroke.

One of the complexities of backstroke is that you're normally doing two things at the same time: a recovery with one arm and a stroke with the other. An innovation Terry introduced in the Backstroke DVD is to simplify this by practicing with only one arm at a time, so you are doing only one arm movement at a time.

Once you've mastered head lead and hand lead sweet spot, the one-arm stroke is just a matter of switching from your head lead sweet spot on one side to your hand lead sweet spot on the other side, and back again. This keeps your focus on using your core body rotation to power your arm movement while maintaining balance and streamlining.

Once you've practiced that on both sides, transitioning to full stroke backstroke is just a matter of combining your arm recovery on one side with your arm stroke on the other, and vice versa.

A little tip to better your backstroke, is to think of using the shoulder and not to think about the arm or hand for arm recovery . Think of shrugging or popping the shoulder up as you begin recovery then and let the shoulder brush lightly against your face as it passes by . It will keep you more in rhythm. The shoulder is an extension of motion that is set up by the hips and core .This tip definitely help my backstroke.

Dave

I used to that till a couple training sessions. I found out that my under water motion was not as effficient as it could and as I was not properly using both the strength coming from the torso rotation as well as my back muscles.
Now I do not pop my shoulder out prior to the recovery I push (/rotate) the appropriate side of my thorax up.
It works better for me, my feel is that I get more leverage from my core muscles and it further fine tunes the hips/thorax/shoulder rotational "syncing" you are describing.